# A line segment has endpoints at (5 ,2 ) and (3 ,4 ). If the line segment is rotated about the origin by (3 pi)/2 , translated vertically by 4 , and reflected about the y-axis, what will the line segment's new endpoints be?

Apr 30, 2017

The new end points are $\left(- 2 , - 5\right)$ and $\left(- 4 , - 3\right)$

#### Explanation:

We are going to use matrices.

The matrix of rotation of $\frac{3}{2} \pi$ about the origin is

$\left(\begin{matrix}0 & 1 \\ - 1 & 0\end{matrix}\right)$

The matrix of reflection in the $y$-axis is

$\left(\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right)$

The combination of the 2 operations is

$\left(\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right) \cdot \left(\begin{matrix}0 & 1 \\ - 1 & 0\end{matrix}\right) = \left(\begin{matrix}0 & - 1 \\ - 1 & 0\end{matrix}\right)$

The new end points are

$\left(\begin{matrix}0 & - 1 \\ - 1 & 0\end{matrix}\right) \left(\begin{matrix}5 \\ 2\end{matrix}\right) = \left(\begin{matrix}- 2 \\ - 5\end{matrix}\right)$

and

$\left(\begin{matrix}0 & - 1 \\ - 1 & 0\end{matrix}\right) \cdot \left(\begin{matrix}3 \\ 4\end{matrix}\right) = \left(\begin{matrix}- 4 \\ - 3\end{matrix}\right)$